Find the Derivative of Each
Calculus : Derivative Applications – Word Problems Name ______
Related Rates Word Problems #2 Date ______Period ______
Solve each.
1. The volume of a cube is decreasing at the rate of 10 cubic meters per hour. How fast is the total
surface area decreasing when the surface area is 54 ?
2. The 1989 Alaskan oil spill created a circular pool whose area increased at the rate of 45 square
meters per minute. How fast was the radius of the pool increasing when the radius was 5 meters?
3. Assume that Jerry and Susan leave North Fulton Stadium from the same point at the same time. If
Jerry runs south at 4 miles per hour and Susan runs west at 3 miles per hour, how fast will the
distance between Jerry and Susan be changing after 10 hours?
4. Katie was at Marist flying a red kite at a constant height of 400 meters. The kite was moving
horizontally at a rate of 30 m/second. How fast must Katie pay out the string when the kite is 500
meters away from her?
5. The company “Stefanie” that creates costumes for high school musicals, hired Mr. Sergio Stadler (a
well-known mathematician) to calculate the revenue R equation and the cost C equation, which are
given below.
Let x represent the number of costumes produced per week. If the rate of production is increasing
by 50 costumes per week, and the present production is 300 costumes per week, find the rate of
increase in:
a. the revenue R b. the cost C c. the profit P
6. A rectangular box with height 30 cm. is changing in such a way that its length is decreasing at the
rate of 3 cm/hour, and its width is increasing at the rate of 2 cm/hour. At what rate is the volume of
the box changing when the length is 70 cm. and the width is 50 cm.?
7. Avi was fishing in Panama City and finally hooked into the “big one”, only to discover it was a
floating log. If the tip of Avi’s rod is 6 ft. above the water, and he is reeling in the line at a rate of 3
ft. per second, how fast is the log moving towards Avi when there is still 10 ft. of line between the
tip of the rod and the hook?
8. A ladder 13 m. long slides down the side of a water tower. At the moment the bottom end is 12 m.
from the water tower, the opposite end of the ladder is sliding down at the rate of 3 m/h.
a. At that instant, how fast is the bottom of the ladder moving away from the tower?
b. At that instant, how fast is the area of the region created between the ladder, the ground, and the water tower changing?
9. Shadow Problem
A 6 foot tall man is walking away from a wall at 4 ft/sec. A light
on top of a light post 40 feet high is casting a shadow the man walks.
When he is 30 feet from the lamp post:
a. how fast is the length of the shadow changing?
b. how fast is the tip of the shadow moving?
10. Cone Problem
A water tank has the shape of an inverted circular cone with a base
radius 2 m and height 5 m. If water is being pumped into the tank
at a rate of 2 , find the rate at which the water level is rising
when the water is 3 m deep.